Notes on the ordered set $A^A$. Part III. Exponent-cancellations

We continue the study of exponent-cancellation for finite ordered sets. It is known that $A$ can be reconstructed from $A^A$, from $(A^A{)}^A$, and from $A^{A^A}$. In this note we prove the next result in this hierarchy: the ordered set $(A^{A^A}{)}^{A^{A^A}}$ determines $A$ up to isomorphism.